stress distribution in and deflection properties of tri
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South Dakota State University South Dakota State University
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Electronic Theses and Dissertations
1967
Stress Distribution in and Deflection Properties of Tri-plate Beams Stress Distribution in and Deflection Properties of Tri-plate Beams
in the Plastic Range in the Plastic Range
Richard Clarkson Cline
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Recommended Citation Recommended Citation Cline, Richard Clarkson, "Stress Distribution in and Deflection Properties of Tri-plate Beams in the Plastic Range" (1967). Electronic Theses and Dissertations. 3286. https://openprairie.sdstate.edu/etd/3286
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.. srREss DISTRIBUTION IN AND DEFLECTION PROPERTIES OF
TRI-PLATE BEAMS IN THE PLApTIC RANGE
BY
RICH.ARB CLARKSON CLINE
A thesis submitted in partial fulfillment ef the requirements for the
degree Master 0f Science, Major in Civil Engineering, Seuth Dakota
State University
SOUTH DAKOTA STATE ·uNJYeRSITY llBRARY
,,
STRESS DISTRIBUTION IN AND DEFLECTION PROPERTIES OF
TRI-PLATE BEAMS IN THE PLASTIC RANGE
This thesis is approved as a creditable and independent
investigation by a candidate for the degree, Master of Science,
and is acceptable as meeting the thesis requirements for this
degree, but without implying that the conclusions reached by the
candidate are necessarily the conclusions of the major department.
Thesis Adviser Date
Head, Civil\_}ngi�ering Department
.. ACKNOWLEDGMENT
The author wishes to express his gratitude to Dr. Zaher Shoukry,
Associate Professor, Department of Civil Engineering, for his wise
counsel and encouragement throughout the course of this project.
Dr. Shoukry's valued assistance and inspiring help is sincerely appre
ciated. To him, the author is truly indebted.
This thesis is dedicated to the author's wife, Diane, whose loyal
encouragement and sacrifice has made this project possible.
RCC
. . TABLE OF CONTENTS
Chapter Pagli}
I. INTRODUCTION • • • • • • • • • • • • • • • • • ·• • • 1
II.
III.
General . . . . . . . . . . . . . . . . . 1
Background. • • • • • • • • • • • • • • • • • 2
Review of Past Investigations . . . . . . . . 5
Object and Sc0pe of Investigation . • • • • • 8
THEORETICAL BEHAVIOR OF TRI-PLATE BEAMS. . . . . . . Introduction.
Pure Bending.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
12
Combined Shear and Bending. • • • • • • • • • 23
Deflection • • • •
LABORATORY VERIFICATION. . . . . . . . . . . . . . . . . . . . . . . . . .
Materials and Test Specimens.
Testing Procedure • • . . . . . . . . . . . . A.
B.
Static Loading·.
.Fatigue Loading.
• • • • • • • • •
. . . . . . . . . Specimen TF-1
Specimen HS-2 p
Specimen TS-3
Specimen TS-4
. . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
26
31
34
34
35
37
41
49
54
. . Chapter Page
IV. TEST RESULTS • • •.• • • 61
61
62
63
65
68
68
68
70
71
74
V. SUMMARY
Specimen TF-1 ..
Specimen HS-2 . Specimen TS-J
Specimen TS-4 .
.
. AND CONCLUSIONS.
Summary • • •
Conclusions • • •
. . • • . .
. • . . . •
. . . . . . . . . .
. . . • . . . . . . . ..
• •
. .
. .
. .
. .
.
.
.
.
e fl e O • e • • e e • e
Recommended Areas of Future Study . . . . NOTATION • •
BIBLIOGRAPHY
APPENDIX A:
. . . . . . . . . . . . . .
. ..
. .
DERIVATION OF MOMENT-CURVATURE RELATIONSHIPS FOR PURE BENDING. • CJ • •
APPENDIX B: DERIVATION OF T:IEORETICAL DEFLECTION
75
EQUATION. • • • . • • • • • • • • • • • 79
APPENDIX C: PUMP READINGS, ACTUAL LOAD, AND ACTUAL MOMENT RELATIONSHIP FOR EACH SPECIMEN . 81
Figure
1.
2.
LIST OF FIGURES
Typical Shear-Moment Diagrams for Test Specimens
Idealized Stress-Strain Diagram • • • • . . . . . . . . . J G Moment-Curvature Relationship for a Tri-Plate Beam
4. Distribution of Strain, Stress, and Yielding at the Upper Limit of Each Stage of Loading • • e •
Page
10
13
14
17
5e Moment-Curvature Relationship for a Hybrid Beam. • 22
6. Distribution of Strain, Stress, and Yielding at the Upper Limit of Each Stage of Loading (Hybrid Beam) • 24
Details of Specimens TF-1, HS-2, TS-J, and TS-4 • • . . . ?.
8.
9.
Test Setup for Static Load Test • • . . . . . . . . . . . Test Setup for Fatigue Test. . . . . . . . . . . . .
10. Electric Timer and Hydraulic Control Apparatus for Fatigue Testing Machine • • • • • • • • • • • •
11.
12.
lJ.
14 ..
15.
General View of Fatigue Test Setup . . . . Strain Gage Arrangement, Specimen HS-2 • • 0 • •
Lateral Support _Device . . . . . . . . . . . . Lateral Supports in Testing Machine. . . . . . . . . . . Typical Specimen Mounted in Testing Machine.
16. Load Versus Deflection at Midspan, Beam HS-2 .
32
36
J8
39
40
42
4J
44
45
48
17. Lateral Buckling of Compression Flange of Specimen HS-2. 50
18. Strain Gage Arrangement, Specimen TS-J 51
19. Load Versus Deflection at Midspan, Beam TS-J 53
20. Lateral Buckling of Compression Flange, Specimen TS-J. • 55
Figure
2L
22.
23.
24.
. .
Strain Gage Arrangement, Specimen TS-4 . • •
Load Versus Deflection at Midspan, Beam TS-4 . . . . . . . Lateral Buckling of Compression Flange of Specimen TS-4.
Typical Yield Lines in Zone A of Specimen TS-4 . . . . .
Page
56
58
59
60
. . LIST OF TABLES
Table
I. Mechanical Properties of Constructional Steels • •
II .. Dimensions and Sectional Properties of Test Beams.
III Q Mechanical Properties of Test Specimens • •
IV. Bending Properties of Test Beams • • • • • •
v. Theoretical Versus Observed Moment . . . .
Page
4
33
35
35
68
General
CHAPTER I
INTRODUCTION
Tri-plate beams combine versatility, economy, and strength in
steel design. These beams have many applications which indicate
increased efficiency and economy in long-span plate girder construction.
Their structural advantages are almost unlimited.
Tri-plate beams are fabricated by welding three different types
of steel to form an I-section. Each flange and the web of the section
is fabricated out of a different yield strength steel. Because the
flanges of a beam contribute most of its moment resistance, the use of
high-strength steel in the flanges is desirable. However, in tri-plate
beams, the tension flange consists of a higher strength steel than does
the compression flange because of the buckling liability of the compres
sion flange. The utilization of a higher strength steel in the tension
flange also makes �bssible a reduction in the tension flange cross sec
tional area. Because of its low moment carrying capacity, the web is
fabricated out of a lower strength steel.
The idea of tri-plate beams is not totally new, having previously
been used to a very limited extent in plate girder construction. How
ever, these beams have not previously been recognized and analyzed as a:r1,
independent part of the girder system. In a way, the tri-plate concept
is analogous to composite design, where the reinforced concrete slab is
used as a large compression flange, the web is of a low strength steel,
and the tension flange is composed of a high�strength steel.
2
High-strength steel beams have indicated increased economy in
steel design e Tri-plate beams will offer still more economy while, at
the same time, provide new possibilities in construction with structural
steel e Furthermore, these beams represent savings in material and the
availability of lighter sections that can meet the competition presented
by prestressed concretea
Background
The rapid increase in the use of prestressed concrete construc
tion has resulted in a search f0r more efficient and economical methods
of construction with structural steel by the steel industry. Starting
about thirty years ago, the steel industry has conducted a continuing
program of research to develop steels of lower cost and higher perfor
mancee As a direct result, a large selection of new economic and high
performance steel is now available to the designer e
The 1963 AISC (American Institute of Steel Construction) speci
fication allows the use of a variety of high-strength and high-strength
low-alloy steels such as the present ASTM-AJ6, A440, A441, and A242
designationso The yield strength of these new steels may vary from
36,000 psi for AJ6 carbon steel to 50,000 psi f0r A4l+l high-strength
low-alloy steel. At present, however, the steel industry is promoting
a new group of higher strength structural steels that offer possible
weight and cost savings over the steels currently covered by the AISC
spe�ificationo These new heat-treated constructional alloy steels have
a yield strength in the vicinity of 100,000 psi o
With these new steels at its disposal, he steel industry began
searching for a structural member that could favorably compete with the
economy and efficiency of prestressed concrete., With their high yield
strength, good ductility, and excellent welding qualities, these newly
developed high-strength steels indicate that they can offer the desired
competitive designG
The structural steels that are widely used at present can be
divided into the following three broad categories:
l o Structural carbon steels.
2� High-s_trength low-alloy steels.
J. Constructional alloy steelsG
Pertinent information concerning the steels that are representa
tive of these three categories is given in Table I .,
Of particular significance in this study are the ASTM-AJ6,
ASTM-A441, and "T-1" designations., AJ6 is a structural carbon steel.
A441 is a low-alley, mangenese-cop�er-vanadium steel intended fer
applications requiring cold forming, welding, and improved toughness
properties . USS "T-1" constructional alloy steels are heat-treated
steels suitable for welded structures.
In contrast with many of the higher strength steels avail
able in the past, the modern high-strength steels show more favor
able price-to-strength ratios than structural carbon steel�l
1Ge Haaijer, "Economy of High Strength Steel Structural Members," Transactions, American Society of Civil Engineers, volo 128, 1963, p. 8200
3
Category
Structural Carbon St_e·el
High-Strength Steel
., �-.... � -C�omu. �loy Steel
Table I
. ,MeohanioaJ.--�aperties 0£ Constructiona1-Steels
, '
Steel
ASTM ... A7 ASTM-A36
ASTM,..A440
.ASTM-A441
USS COR-TEN
USS "T-1'"
Thickne.ss Range , (inches)
.All thicknesses 4 and und:er
,-.;,
3/4 and under over 3/ 4 to 1,.'"# 2 incl •. over 1 1/ 2 to 4 incl.
'
J/4---md"unde,r over 3/ 4 to 1 1/ 2 incl. over 1 1/2 to 4 incl.
1/2 and under over 1/2 to 1 1/2 incl. over 1 1/2 to 3 incl.
3/.16 ;o 2 1/2 ·incl.. over 2 1/? to 4 incl.
Yield Point (psi)
33,000 36,000
59,000 46,ooo 42,000
50,000 46,000 42,060
50,000 47,000 43,000
1.00,000 90,000
- '
Tensile Strength
(psi)
60,000 te 75,000
70,000 min. 67,000 min. 63,000 min.
70�000 min. 67 ,000:. mi» .. 63,000 min.
70,000 min. 67,0.00 min.
- 63,000 min.,
115,000 to 135,000 10.5,000 to 135,000
�
That is si high-strength steels show a relative increase in price that is
les.� ·han the relative increase in yield -strength_. As a result, the
applica'.ion of higher strength steels to structures in which the higher
streng h can be utilized often results in significant material-cost
saving 'o
Since the moment resistance pr0vided by a beam or girder is
contr:ibuted mostly by the flanges, replacing the flanges with higher
strength steel results in considerable saving in cost as well as in
wei.gh " Cost and weight analyses have indicated that beams constructed
with high-strength steel flanges have resulted in as much as 20 per ent
saving :i:n c st and as much as 40 percent reduction in weight ,. Since
he web )f a plate girder accounts for a large portio of its weight
but c n+�ributes only a small part of its moment carrying capacity, the
use of -�he.aper low-strength steel for the web would be desirable.
The proper selection of material is one of the essential steps
in a design to ensure that the structure will meet its functional
requirements with adequate safety and minimum cost ., The question of
economics :includes the base price 0f the material, fabrication costs,
5
fr ·igh-, effect of dead weight of the structure n foundation costs,
optimum space utilization, and some other factors., With these consider-.
ations in mind, the tri-plate beam will be a significant contribution in
th area of steel construction.
Review f Past Investigations
Al hough the use of girders with carbon steel webs and higher
s rength flanges of structural silicon steel dates back many years, the
theoretical behavior of such members has not been analyzed extensively,
and little experimental work has been performed to verify the theoreti
cal behavior .
The availability of the new high-strength structural steels has
prompted the recent investigations of the behavior of hybrid beams that
are built up by welding higher strength steel flanges to lower strength
steel webs ., The results of studies presented by Haaij'er in 19612 indi
cate that thr0ugh the optimum design of structural members, the appli
cation of high-strength steels can lead to lighter weight ·structures
and often to significant.material-cost savings.
Tests conducted in 1961 by A . A . Toprac at the University of
Texas yielded the following conclusions: 3
1. When a girder with high-strength steel flanges and carbon
steel web is loaded s0 that the yield point of the web is
exceeded, a redistribution 0f stress will occur and allow
the girder to continue to deform elastically . When, after
having the load rem0ved, the girder is rel0aded, cemplete
elastic behavior extends over a load range almost twice
the first range.
2Ibid .
JA ., A . Toprac and R . A . Engler, "Plate Girders With High Strength Steel Flanges and Carben Steel Webs, " Proceedings, American Institute of Steel Construction, 1961, p . 91 .
6
7
2. Girders of this type are particularly adaptable to long spans
where the ratio of dead-to total load is high and live load
deflection is not critical.
J. The fact that the web is stressed beyond its yield strength
has litt1e adverse effect on the ultimate carrying capacity
of the member.
More recent theoretical and experimental investigations of the
behavior of hybrid beams have provided new and valuable information
concerning the design of such beams. As a result of investigations
made in 1964, Frost and Schilling4 have concluded that because the
reduction in bending strength caused by shear in hybrid beams is small,
it is satisfactory to design hybrid beams independently for bending and
shear. In 1966, tests conducted by Sarsam5 showed that thin webs can
be used satisfactorily in hybrid beams, especially for laterally sup
ported, long spans. Also pointed out was the fact that the stability
rather than the stresses will control the design for thin web hybrid
beams because of the critical lateral and torsional buckling of the
compression flange.
4R, W. Frost and C. G. Schilling, "Behavior of Hybrid Bearn.s Subjected to Static Loads," Journal of the Structural Division, ASCE, vol. 90, no. STJ, Proc. Paper 3928, 1964, p� 86.
5,Ja:ma:1 · :S. 61 Sarsam, "Behavior of Thin Web Hybrid Beams Subjected to Static �Loads-," thesis presented to South Dakota State ·university, Brookings, South Dakota, in August, 1965, in partial fulfillment of the requirements for the degree of Master of Science, p. 63.
The need for further testing of beams fabricated from high
strength steel is evident. The study of tri-plate beams can furnish
additional insight into the behavior of such sections and provide for
new innovations in structural steel design.
Object and Scope of Investigation
The selection of A441 high-strength low-alloy steel for the
compression flanges and AJ6 carbon steel for the webs was made because
of the increasing demand and popularity of these two types of steel .
In 1963, with its favorable price-to-strength ratio and excellent weld
ability,'A441 steel was used in 87 percent of the buildings fabricated
from high-strength steel .
In this investigation, "T-1" constructional alloy steel has
been used for the tension flanges . Although it is not yet cevered by
the AISC specification, it has nevertheless been chosen because of its
high-strength properties. This use of "T-1" steel will also provide
additional information about its structural behavior, weldability, and
adaptability for future building construction .
The objectives of the investigation reported herein are as
follows: ·
1 . To investigate the structural behavior of tri-plate beams,
both in the elastic and plastic range .
2 . To study the stress distribution in the webs and flanges of
tri-plate beams .
J. To determine their strength and mode of failure.
8
4 o To present a simplified method of stress and deflection
analysis and compare calculated values to observed strengths
and deflections .
5. To investigate the fatigue properties of tri-plate beams .
This study has been limited to pure bending and to combined
shear and bending . These two cases are illustrated in Figure 1 .
Zone A is pure bending; Zones B and Care combined shear:,and bending .
The plastic design method of analysis has been used in the theoretical
considerations to obtain the practical upper limit of strength of the
beams .
9
p
,, X
I C
..
p p
• �
I I
L
I I I
sbear Diagram) I I
p
I
�I I I
I
I
I
fume C fume B Zone A It I • • IC
I
Zone B ., i • Zone C �
Bending Moment Diagram.
Figure 1. Typical Shear-Moment Diagrams for Test Specimens
10
CHAPTER II
THEORETICAL BEHAVIOR OF TRI ... PLATE BEAMS
Introduction
The behavior of tri-plate beams in pure bending and combined
shear and bending will be investigated under static loads. The load
deflection or moment-curvature rele:tic>-nship, and the progressive stages
of yielding from the initial yield to the fully plastic condition will
be epecilically analyzed and compared with hybrid beams.
Figure 1 shows the shear and moment distribution resulting from
the loads P on the test specimens. The values for the shear and moment
at the different z ones along the beam are expressed as followe:
where
Zone A:
Zone B:
Zone C:
M = 3PL 5
V = 0
- L) M = P(¾ + 5 V = p
M = 2Px 0
V = 2P
P = the oonoentrat•d load
L = the length of apan
where
x = the distance from the support to any point in the C
corresponding zone
In the subsequent analysis, the following assumptions are used:
1 . The stress-strain relationship for all parts of a tri-plate
beam can be idealized to consist of the two straight lines
shown in Figure 2 . The �quations of these two lines are
O"'" = EE
r::T = fT y
(elastic range)
(plastic range)
tr= the stress at .the extreme fibers of the section
cry= the yield stress at the extreme fibers of the section
E = the modulus of elasticity of steel
f = the elastic strain
12
2. Deformations are sufficiently small so that tan (tangent) ¢
can be considered equal to¢ except for large values of¢
that occur in the fully plastic range . (¢ is the curvature)
Pure Bending
The analysis of a tri-plate beam can best be explained by means
of a bending moment versus curvature diagram as that shown in Figure J.
As the moment increases, the moment-curvature curve progresses through
six stages of yielding . Within Stage I and up to the yield moment of
the bottom web fibers, M , the stress distribution is linear and varies Y1
according to the flexural formula:
(T = � I
b .. U) U)
Q) H �
Elastic Plastic �-R�a_n_g_e---r-----------oange
199640
er-= v::
Strain, t
Figure 2. Idealiz ed Stress-Strain Diagram
SOUTH DAKOTA STATE ·uNJYeRSITY LIBRARY
13
� .. +,) s:: � 0
� bD s::
•rl s:: Q)
P'.:l
M �- ____ -V-�.o------V.;..;;I;;.._ _____ �
2-4-----·
ct t
C
Curvature, ¢
Figure J. Moment-Curvature Relationship for a Tri-Plate Beam
14
d
where
r:T= the stress at the plane under consideration
M = the resisting moment of the section
15
y = the distance from the neutral axis to the plane 0f stresses
I= the moment of inertia of the seC'tion
Stage I represents the range in which the beam is fully elastic .
The moment is directly prop0rtional to the curvature and the following
relations for elastic bending are valid:
From which
and
where
M = EI¢
M y
¢=the curvature
= taI y
J> = the radius of curvature
E = the modulus of elasticity 0f steel
(Ty= the yield stresses at the extreme fibers
The initial yielding of the tri-plate beam takes place only in
the bottom fibers of the web. Since the extreme bottom fibers of the
web are at a distance cb from the neutral axis, formula (3) becomes
(1)
(2)
(3)
(4)
where
M = the lower web yield moment Y1
V-yw = the yield stress of the web
cb = the distance from. the neutral axis to the extreme bottom.
web fibers
16
As the moment increases beyond the upper limit of Stage I, the
M-� curve passes through five other stages. Figure 4 shows the devel
opm.ent of strain, stress, and yield distribution as the tri-plate beam.
is bent in successive stages beyond the elastic limit (Stage I) and up
to the plastic limit (Stage VI) .
Stage II represents the range in which yielding develops in the
extreme bottom. fibers of the web, while the extreme top web fibers and
the flanges remain elastic, as shown in Figure 4. In this stage as well
as in Stage I, the strain varies linearly through the depth of the cross
section; however, in Stage II, the stress is no longer directly propor
tional to the strain and the stress distribution is as shown in Figure 4.
The moment at any given stage can be obtained by integrating the stress
areas according to the general expression
M = (vd.Ay �:ea
Thus for Stage II, the moment corresponding to a given curvature can be
obtained by integrating the first moment of the stress over the entire
beam. cross section, which gives
(5)
�!
E4),
Stage I
-+f '93 � �f �00 I
�
:::1 ,_. --
� �3 r-
f-- '=� .. � Strain
-.f �3 I-
��� �
-.,�3 I-- -I Vy3 I--
r-�.-., t- v;4 � I-- IT��
Stage II Stage III
Str,:ess
Yielding
Stage IV Stage V
Figure 4. Distribution of Strain, Stress, and Yielding at the Upper Limit of Each Stage of Loading
Stage VI
f-J ---:1
where
Ift = the moment of inertia of the top flange about the neutral
axis
Ifb = the moment of inertia of the bottom. flange about the
neutral axis
t = the web thickness w
The integration details of the derivation are given in Appendix A.
Considering that the maximum elastic strain for Stage II is given by
u: E-= .:J::!!.. E
and tan¢
Then
18
(6)
where
ct= the distance from the neutral axis to the extreme top web
fibers V:
By replacing E¢ in equation (5) with�, the value of the moment at ct the upper limit of Stage II is given by
I I t c 2
M = fl: ( ft + fb)+
� ( w b
Y2 yw ct yw 2
(7)
In Stage III yielding develops in the extreme top web fibers
while the flanges are still elastic. By integrating the stress areas,
the moment corresponding to a given curvature is
t C7: 3 w yw
JE2ql (8)
where
Z = the plastic modulus of the web w
19
The upper limit of this stage is the m.om.ent that initiates yielding in
the upper flange; the strain is then related to the curvature by
V'y 3
and tan�=�
From. which
where
� = the yield stress of the upper flange 3
dt = the distance from. the neutral axis to the extreme top
flange fibers
(9)
Since equation (9) is the curvature relationship at the upper limit of
Stage III, equation (8) becomes
Ift + rf· b) + M = V:- (--- � Z Y3 YJ
dt yw w
t d 2 w t (10)
In Stages IV, V, and VI the behavior of the beam. is similar to
that in Stages II and III, and the mom.ant corresponding to a given cur
vature can be obtained from. the stress distribution for that curvature.
Stage IV represents the range in which yielding progresses through the
upper flange. The M-� relationship for this stage is
+ v- z yw w ( 11)
20
where
Zft = the plastic modulus of the top flange
The curvature relationship at the upper limit of this stage (initial
yielding of the lower flange) is given by
Then equation (11) becomes
+ tr z yw w
(12)
(13)
Stage V represents the range in which yielding progresses through
the lower flange. In this stage, as in the other stages, the :strain
varies linearly through the depth of the cross section, however, in
Stage V the strain exceeds the elastic strain of both flanges. In this
stage m.ost of the beam. is in the plastic condition except a small por
tion around the neutral axis, as shown in Figure 4.
In Stage VI the remaining elastic portion of the web at the
neutral axis becomes plastic, and the entire section is in the plastic
condition. In the plastic range the moment-curvature relationship and
the magnitude of the maximum. plastic moment are similarly derived by
considering the deformed structure and obtaining the corresponding cur
vature and moment. The M-� relationship for Stage VI is determined as
(14)
where
Zfb = the plastic modulus of the bottom. flange
21
Frost and Schilling recornmend6
that¢ in equation (14) should be
replaced by tan ¢ so that as¢ approaches �/2, the moment approaches
the fully plastic moment, M . Considering that the tan 'Tr/2 is unde-
fined, equation (14) becomes
M = fJ: z + � zft + o: zfb p yw w YJ Y4_
It can be seen from. equation (15) that the expression for the
plastic moment, M , is merely an application of the general plastic p
moment equation
M = V: Z p . y
(15)
to each part of the beam. By multiplying the plastic modulus of each
part of the beam. by its respective yield stress, the plastic moment is
obtained.
where
t ; ( ct
2 + cb
2) = the plastic modulus of the web
tf bt tf (ct + 2) = the plastic modulus of the upper flange
tf bb tf (cb + 2) = the plastic modulus of the lower flange
The typical m0m.ent-curvature relationship of a hybrid beam indi
cates that, as the moment in.creases, the M-¢ curve passes through four
stages of yielding, as shown in Figure 5. The distribution of strain,
6Frost and Schilling, p. 60.
M IV L - - - --,-ei------------------� M f III
n_ __
Curvature, (1)
Figure 5. Moment-Curvature Relationship for a Hybrid Beam
22
23
stress, and yielding at the upper limit of each of these stages is shown
in Figure 6. Whereas the extreme top and bottom fibers of the web of
the hybrid beam begin yielding simultaneously at the upper limit of
Stage I , the initial yielding of the tri-plate beam takes place only in
the bottom. fibers of the web. Also , the initial yielding in the web of
a tri-plate beam. begins at a lower moment value than it does in a hybrid
beam .
Combined Shear and Be�ding
The effect of shear force on a tri-plate beam may limit the load
to a value that is less than that which corresponds to the full plastic
moment. This effect is especially true in the case of a short beam cen
trally loaded or a long beam with a concentrated load near the end .
In order to arrive at a basis for assuring that the full plastic
moment will be reached , two possibilities must be considered:
1. General yielding of the web because of shear may occur in the
presence of high moment-to-shear ratios .
2. After the beam has become partially plastic at a critical
section because of flexural yielding , the intensity of shear
stress at the centerline may reach the yield condition .
In cases of high moment-to-shear ratios (Case I) , it can be
assumed that the entire moment is taken by the flanges and that the
shear is resisted by the web.
V , = A w
(17)
24
. ,., I E ---00
I Stage I Stage II
Strain
Stress
Yielding
Stage III
______, -------
I-
I Stage IV
Figure 6� Distribution of Strain, Stress, and Yielding at the Upper Limit of Each Stage of Loading
Hybrid Beam
where
!= the shear stress in the web
V = the applied shear force
A = the cross sectional area of the web w
25
From. equation (17), the maxim.um. possible shear according to this cri�
terion is given by
where
V = the p
'y = the
t = the w
C = the
V =V =7":tc max p y w
shear force that causes the web
shear yield stress of the web
thickness of the web
depth of the beam.
to become fully plastic
Using the Von Mises-Hencky7yield criterion for yielding under pure
shear,
the plastic shear force is given by
t7: V = ....::...1lJ!.. t C p {J w
(18)
(19)
Because that part of the web that yields in moment does not con
tribute to shear resistance, the shear stresses must be carried by the
remaining elastic portion of the web around the neutral axis (Case 2).
For any given stage of yielding, the appropriate relationship between
7Joint Committee of the Welding Research Council and the American Society of Civil Engineers, Commentary on PlastJ..c Design in Steel, 1961, p. 37.
the plastic moment, M , as modified by shear and the full plastic ps
moment, M , can be determined by consideration of the M-¢ relationship p
26
and the shear stress distribution for that stage. However, the implied
reduction in moment capacity does not actually occur because of strain
hardening.
Because of the stress concentrated at these points of inter
action, strain-hardening is possible. Interaction between bending and
shear occurs only when both types of stresses simultaneously reach high
values. Since high shear and moment values occur in very localized
zones of steep moment gradient, · the beneficial aspects of strain
hardening usually enable the moment to reach the full plastic value.
Therefore, in a tri-plate beam, no reduction in the plastic moment, M , p
is necessary if the value of' the shear at the ultimate load does not
exceed V , as given by equation (19). p
The presence of shear has both a detrimental and a beneficial
aspect. The beneficial a_spect is due to the fact that shear forces
always imply a moment gradient and, therefore, only a short girder
portion is affected by the maximum. moment. The adverse aspect is that
a web which has yielded in shear cannot take its allotted bending moment
and the flanges will have to compensate for it, resulting in excessive
deflection and loss of load carrying capacity of the beam.
Deflection
Although strength to carry the load is the primary matter of
importance in most structures, certain problems related to deflections
can be of major concern. The deflection may exceed established limits
set up by a ' building code or specification. Excessive deflections may
hinder the operation of moving parts or they may cause the cracking of
plaster in finished ceilings. Thus, it is desirable ' to establish a
method for calculating the deflections for a tri-plate beam and not to
exceed recommended values established by existing codes.
27
In the elastic range, deflection of a flexural · m.ember varies
inversely with the moment of inertia. In an I-shaped member the flanges
account for most of the moment of inertia since they are farthest fr0m
the neutral axis. When high strength steel is used for the flanges,
their area is reduced to such an extent that the moment of inertia of
the section is reduced to a value that may be inadequate for deflection
purposes.
The theoretical deflection at the center: line of a beam l0aded by
symmetrical four-point loading is
(20)
where
� = the centerline deflection in inches
p = the load
L = the span length in inches
E = the modulus of elasticity of steel in psi
I = the moment of inertia 0f the section in inches to the f0urth
power
For the details of the derivation of equation (20) see Appendix B.
The use of a relatively small moment of inertia in this formula
might result in excessive deflecti�ns.. Specificati,ms usually establish
28
a certain minimum depth-to-span ratio which serves to control deflec
tion.
For a given uniformly distributed live load, the deflection of a
beam is
where
K1 = a constant incorporating the end restraint
wL
= the uniformly distributed live load
The total load, including the · effect of . live load, creates· a bending
stress of
where
K2 = a constant of end restraint
wT = the total uniformly distributed load
d - the depth of the beam
Combining equations (21) and ( 22) to. eliminate I gives
f! - 2 Kl wL rT .1._
L - K2 WT E A
(21)
(22)
(2J )
Consideration of the high level of stres,s developed in tri-plate
beams and the deflection limitations established by various specifica
tions indicates that equation ( 2J) will yield relatively large span-to
depth ratios ·ror high strength steel . Large d/L ratios require rather
deep girders. However, if the live load is considered to be 50 per.cent
of the total load, the span-to-depth ratios will be reduced to values
which are in line with present practice; i. e. , smaller depths can be
considered. 8
Deflections of members in the plastic range can be obtained by
integration of the basic differential equation:
2
29
s!iC = ¢ dx2 ( 24)
where
x = the distance along the member
y =
d 2 � -dx2 -
� =
the deflection from the original straight line of the member
the approximate curvature of the member
the curvature caused by bending
The use of equation (24) in the plastic range depends on the
assumptions made in the moment-curvature relationship and on the methods
of handling boundary conditions.
The problem of calculating deflections for tri-plate beams might
be generalized by the following statement:
In spite of the differences in the design basis, the working loads for both plastically designed beams and those designed by the allowable stress procedure are usually in the elastic range. Consequently, the c,.lcula9ion of deflection at working load is in general identical.
The working load for tri-plate beams will be approximately halfway
between the upper web yield moment and the upper flange yield moment in
8 Toprac and Engler, p. 91 . 9L . S . Beedle and Associates, Structural Steel Desigri, The Ronald
Press Company, New York, 1964, p. 222.
30
Stage III . The deviation of the moment curvature curve from a straight
line in Stage III is almpst negligible until the moment approaches the
yield moment of the flanges. This slight deviation indicates that
yielding of the extreme fibers of the web does not greatly affect the
,curvature. This result is not surprising since the flanges are still
under elastic conditions, and they control the curvature of the section.
Consequently, the deflection of tri-plate beams at working loads can be
calculated on the basis of elastic deflection theory.
CHAPTER III
LABORATORY VERIFICATION
M�terials and Test Specimens
In order to provide an experimental check of the analysis, four
lieams were tested in the Civil Engineering Laboratories at South Dakota
State University, Brookings, South Dakota. The beams were designed to
fit, with the fewest modifications, the available testing equipment.
Ease of handling was also a consideration.
The webs of all four beams were 1/4 inches by 13 inches A36
structural carbon steel. The top flanges of all beams were 4 inches
wide by J/8 inches thiek A441 high-strength steel. The bottom. flanges
of beams TF-1, TS-3, and TS-4 were 2 inches wide by J/8 inches thick
"T-1" constructional alloy steel as shown in Figure ?. The bottom
flange of beam. HS-2, however, was 4 inches wide by 3/8 inches thick
A441 steel, i.e. , beam. HS-2 was a hybrid beam.. These dimensions and
the distribution of material offer a comparison between the results
obtained for the tri-plate beams and those of a hybrid beam.. Table II
gives the cross sectional dimensions and properties for each beam..
The overall length of all four beams was 20 feet with a sps.n
length for each beam. of 19 feet. The beams were loaded as shown in
Figure 1 to simulate, as much as possible, uniform. loading conditions.
No stiffeners were provided at either points of loading or at points of
support.
Eight-inch, gage-length tension coupons were tested to determine
the longitudinal mechanical properties of the plates that were used to
4 ''
� r:.·I �:r � NI
r ���0q
I - � '
rr� �� - - - - - - - -- - - - - - - -- - -- :-... - -- - ---- - ---- - -- -- -- - --- - - - - - - . � - - - - - - - - - - - - - - - - - - - - - - - - - - - - N V . - - - - - - - - - - - - -- - - - - --
re 4" X 3/8" X 20 ' -0" TYPICAL .ALL BEAMS (A441)
\ �
3/ H.
3/ \ (o
PLAN
ao� o'1
--"'"' j - � 13" X 1/ 4" X 20 ' -0" �
I TYPICAL ALL BEAMS (AJ6)
�------- I ELEVATION � 2 11
X 3/8" X 20 '-0" BEAMS TF-1 , TS-J , TS-4 , ( T-1) ft. 4" X J/ 8" X 20 ' -0" Ei\TD -
7 4" -Beam HS-2
VIEW BEAM HS-2 (A441)
Figure 7. Details of Specimens TF-1 , HS-2 , TS-3 , and TS-4
vJ N
Beam TF-1 HS-2 TS-3 TS-4
Beam TF-1 HS-2 TS-3 TS-4
Beam TF-1 HS-2 TS-3 TS-4
Table II
Dimensions and Sectional Properties of Test Beams
r + t f
I I
� Tl
t dt ct -. ,__ w
I , C d
Cb ¾
' ,, I , I I
� bb
tf t bb bt inches inches inches inches
0 0 378 0. 262 1 . 987 3 . 985 0 . 377 0 . 254 4. 004 4 .. 004 0 . 379 0 0 260 1. 993 3 ,, 990 0. 380 0. 258 1. 989 3 . 994
Cb
ct d �
inches inches inches inches 7 . 399 5.615 13. 770 7 . 777 6 . 472 6. 472 13. 699 6. 849 7 . 402 5. 609 13 . 769 7. 781 7 . 403 5 ., 593 13 . 756 7. 783
A I
inches 2 inches 4 z 2fb w
inches3 inches3
5 0 666 142 . 039 11 . 302 8. 741 6. 306 179. 829 10 . 725 10. 050 5 . 650 142. 075 11 . 213 8. 767 5. 626 141 0 541 11 . 105 8 . 773
33
C
inches 13. 014 12. 945 13 . 011 12 . 996
dt inches 5 . 993 6. 849 5. 988 5. 973
zt inches3
5. 699 10. 050
5 . 731 5. 740
34
fabricate the test beams. The tension coupons were prepared from plate
samples furnished by the fabricator. The averages of the results of the
tension tests are listed in Table III . The average yield point for the
web material of all beams was found to be 38, 700 psi. The average yield
point for the compression flange material, as well as the tension flange
of beam HS-2, was 54, 500 psi. The average yield point of the tension
flange material of beams TF-1, TS-3, and TS-4 was found to be 109, 300
psi . The bending properties of each beam, as calculated from the theo
retical equations, are shown in Table IV.
Except at the ends, there· was no flame cutting for all specimens.
The flange to web connections were accomplished by a continuous partial
penetration fillet weld using low-hydrogen electrodes, after the weld
surfaces were mill surfaced.
Testing Procedure
A. Static Loading
All beams were simply supported six inches from each end. The
static loading was applied at the fifth points of the span by four equal
loads as shown in Figure 8. These loads were applied by using a single
60-ton hydraulic jack, located at the bottom of the testing machine and
pulling the assembly through a high-strength steel bar . The dial on the
manually operated pump read the psi pressure in the line. In order to
interpolate the exact reading, the jack was calibrated by proving rings
with the load . applied by a compression testing machine . A jack factor
of 11. 4 was determined. The final preparation before testing was to
Speoim.en
TF-1
HS-2
TS-3
TS-4
Specimen
HS-2
TS-3
TS-4
Table III
Mechanical Properties of Test Specimens Yield Stres s , Psi
Web Compres sion Flimge
38 , 700 54 , 500
38 , 700 54 , 500
38 , 700 54 , 500
38 , 700 54 , 500
Table IV
Bendin� Propertie s of Test Beams
M M M M Y1 ft-�fps
y Y4 ft-kips ft-k�ps ft-kips
83 . 065 83 . 065 106 . 888 1 06 .888
55 .479 75 . 919 94 . 567 1 13 . 104
55 . 263 75 . 798 94. 071 1 12 . 906
35
Tension Flange
109 , 300
54, 500
109 , 300
109 , 300
M p ft-kip�
1 15 . 278
1 16 .292
1 16 . 068
Loading Frame
Specimen
12 ' -0"
1 I
Points of Lateral Support
l-"---60-Ton Hydraulic Jac
l 16'-0"
Figure 8. Test Setup for Static Load Tests
\..,0 °'
paint the specimens with lime whitewash in order to detect zones of
yielding.
B. Fatigue Loading
37
The fatigue specimen was simply supported six inches from each
end. The loading was applied at midspan by an automatically controlled
hydraulic pump as shown schematically in Figure 9 . The electrical con
trol apparatus shown in Figure 10 controlled the rate of load applica
tion on the beam. Lateral bracing was provided at the points of support
only. The test setup was as shown in Figure 11.
Specimen TF-1
Testing started with a 7,000 pound midspan load applied at the
rate of eight cycles per minute; however, the load was later changed to
6,000 pounds so that the capacity of the apparatus would not be exceeded.
A rate of loading greater than eight cycles per minute could not be
maintained because of the limitations of the electrical control device.
Strain gages were not provided for the fatigue testing.
The loading was carried on for varying periods of time with
inte�mittent rest periods . Usually, testing continued for periods of
15 to 19 hours with rest periods between each series of load applica
tions. However, at one point in the testing, the beam was idle for a
period of several weeks because of equipment failure.
Investigations of the weld areas and checks on the deflection
were conducted daily. A centerline deflection of 0 . 05 inches was
Electric Tim.er and Hydraulic Pump
Hydraulic Ram.
.-
0
19 ' -0"
Figure 9. Test Setup for Fatigue Test
Specimen
vJ (X)
Figure 10. Electric Tim.er and Hydraulic Control Apparatus for Fatigue Testing Machine
39
Figure 11 . General View of Fatigue Te st Setup +=" 0
determined during the early loading and showed no change during the
duration of the test.
41
Because of difficulties inherent in the testing equipment, . the
test was terminated after 232, 067 cycles of loading had been applied to
the beam. At this point, there was no indication of any impending fail-
ure .,
Specimen HS-2
Beam HS-2 was the first specimen to be tested under static loads .
Twelve SR-4 type A-1 strain gages were mounted on the beam.. Figure 12
shows the location of these strain gages. All the strain gages were
located at the center of the loading panels . The strain gages on the
web were located two inches from. either the top or bottom flanges .
Strain gage number 5 was located at the neutral axis and strain gage
number J was mounted on the opposite side of the web to give a running
check of the readings at this point ..
To prevent buckling before reaching the plastic condition, the
beam was laterally supported at three points as shown in Figure 8 . The
apparatus used for lateral support is shown in Figure lJ and can be seen
in the testing machine in Figure 14 .
Figure 15 shows a general view of the . test set up for this beam..
The points of bearing between the lateral supports and the test specimen
were heavily greased to minimize fricti0n as the beam deflected verti
cally. The ends of the beam over the support had a clamp-type lateral
support, as shown in Figure 15. Vertical deflection was measured by
placing an Ames dial along the center-line of the beam under the bottom.
I
l
8
- 9
-¥-I, - lo
Figure 12 . Strain Gage Arrangement Specimen HS-2
II
- 12
.{::" N
Figure lJ. Lateral Support Device +=" '-...,J
46
flange . Lateral deflection was also measured by m.eans of 41._1 ind.ice.�
tors located at intijrvals along the upper part of the web .
Testing started by loading the test specimen with a 5,000 fjound
load and unloading without taking any readings of the strain gages or
of the deflection dials . This loadin� was repeated three tim.es . The
reason for this loading and unloading was to release any residual
stresses that may have resulted during fabrication of the beam, No
attem.pt was made to m.easure the existing residual stresses , as suming
that the loading-unloading process would virtually eliminate their
effect .
The beam. was then loaded to a moment of 42 . 123 ft-kips which
corresponded to 14, 780 pounds of actual loading . This mom.ent and load
include the weight of the · beam. and the · weight of the assembly hanging
on the beam.. The pump dial readings and the corresponding loads and
m.om.ents for each beam are given in Appendix C . Because of the weight
of the aseembly, the deflection curves do not start from. zero load,
The load was applied at 2 , 280 pound intervals . At the end of every
interval the loading was stopped and readings were recorded for the
strain gages and deflection dials . The lateral supports were also
frequently checked to see 1£ they were functioning properly. When a
m,om.ent of 42 . 123 ft-kips was reached, a final reading was taken and
the beam was unloaded. During this cycle of loading, the mom.ent applied
was oon.siderably below the web yield mom.ent , � = 83 . 065 tt-kips ; and
the b am behaved elastioally, as expeoted. The train gage readings for
this f1ret cycle o! loading show d very close agr ement with the linear
strain distribution along the cross seotion.
47
The second cycle of loading was to load the beam. beyond the web
yield limit but short of the flange yield moment of 106. 888 ft-kips, ·
then to unload the beam. and observe the amount of residual deflection
resulting from yielding of the web. The load was applied at intervals
of 2, 280 pounds and readings were again taken at the end of each inter
val. The loading was continued until a montent of 94 . 107 ft-kips was
reached, which corresponded to an actual load of 33, 020 pounds. The
beam. was unloaded from the 33, 020 pounds load and was noticed to behave
elastically during the unloading . The vertical deflection dial read
0. 167 inches when the load dial reading had returned to zero as shown
in Figure 16.
A third cycle was attempted with 4, ,560 pound load intervals until
a load of 23, 900 pounds was reached . The load interval was then changed
to 2,280 pounds. The loading was terminated when the moment on the beam
slightly exceeded the flange yield limit of_ 106. 888 ft-kips, . . but, was
below the theoretical plastic moment, M = 11.5 . 278 ft-kips. At this p
point, flaking of the whitewash became apparent on the upper web and
flange. Inspection of the lateral supports showed them to be function� i
ing properly . Since there was no apparent bu6kling in the beam, it was
decided not to load any further but rather to unload the beam. and
observe the behavior of the beam during the unloading cycle. During the
unloading the beam behaved elastically; when the final· load·,·r,eading
showed zero, the amount of residual deflection was 0. 269 inches.
A f0urth and final cycle of loading was tried with the purpose of
observing the maxim.um load carrying capacity of the beam. The beam was
12.8 .. 2
1/4.CJ
99.B
85. ,
·rl 7/.2 �
"'
&j7. 0
4�.8
28. 5
14. 2
o.o
48
40 M f _]f._ - - - - - -
:,o
�: ls •rl ' ' , �
3 ·\·20
1;
io
s o observed
o .._ ___ _._ _______________ �....._---� .<. O ,2. ; /. 0
Centerline Deflection
Figure 16. Load Versus Deflection at Midspan Beam. HS-2
49
loaded "as in the previous cycle with readings taken at the end of each
interval. Until a load of 42, 140 pounds was reached, there appeared to
be a few signs of buckling. However, as soon as the load exceeded
42, 140 pounds, the deflection increased rapidly at a steady rate accom
panied by a drop in load and lateral buckling of the compression flange
appeared as shown in Figure 17. The highest load reached was 42,710
pounds. Figure 16 shows all test results for this beam.
Specimen TS-3
Fourteen strain gages were mounted on beam TS-3, as shown in Fig
ure 18. Strain gage number 11 was placed at the neutral axis, and gages
4 and 9 were mounted on the opposite side of the web. Since the lateral
support devices had functioned properly throughout the first test, no
alterations of design were necessary. The test procedure was very simi
lar to that of Specimen HS-2. The beam was whitewashed and loaded with
5, 000 pounds and unloaded several times to release any residual stresses.
The purpose of the first cycle of loading was to load the beam up
to, but not to exceed, the lower web yield moment of 55. 479 ft-kips.
The beam was actually loaded to a moment of 54. 834 ft-kips at intervals
of 2, 280 pounds. At the end of every interval the lateral supports were
checked and the strain gage and deflection readings were recorded. When
a m-0ment of 54. 834 ft-kips was reached, final readings were recorded and
the beam was unloaded. As expected, the specimen behaved completely
elastic during the loading and unloading of the first cycle.
The second cycle started with the same load intervals and the
same procedure of reading the gages as the first cycle. The intention
Figure 17 . Lateral Buckling of Compression Flange of Specimen HS-2
50
4 \
I \ -- 2
'
9 \
3 ' \ 8 '·
\ .. •/0 '- � _ H
---r -=- " ... � 1:e
7 /.3
l
Figure 18 . Strain Gage Arrangement Specimen TS-3
r
�
V\ t-J
52
of this cycle was to load the beam. to a moment in excess of the web
yield moment . However, as has been previously pointed out, the extreme
fibers at the top and bottom. of the web of the tri-plate beam. do -not
begin yielding simultaneously. Yielding develops first in the bottom.
fibers with yielding in the extreme top fibers t.aking place at a. higher
level of stress. The loading continued until a moment of 61. 332 ft-kips
was reached. This moment exceeded the theoretical yield limit of the
bottom fibers of the web, M = 55. 479 ft-kips . At this point, it was Y1
decided not to unload the beam. but rather to continue loading until the
yield limit of the upper web fibers, M = 75. 919 ft-kips, had been Y2
exceeded . The maximum. moment attained at the end of this cycle was
80 . 826 ft-kips o Before unloading, a slight lateral deflection was indi
cated by the horizontal dials. There was also some evidence of white
wash flaking on the compression portion of the web. When the load had
been completely released, the vertical deflection dial showed a residual
deflection of 0. 175 inche� .
A third cycle of loading was started with intervals of 4,560
pounds. Strain gage readings were recorded as in the previous cycles.
At a load of 19,240 pounds, the rate of applying the load was changed to
2,280 pounds. It was noticed that the beam. behaved elastically and
followed a straig4t line pattern that very closely coincided with the
unloading of the second cycle. The path deviated from the straight-line
pattern when the moment approached the theoretical plastic moment, as
shown in Figure 19. When the moment on the beam. reached 113. 316 ft-kips,
the horizontal dials indicated an excessive lateral deflection; it was
128. 2 4;
//4. 0 40
99. 8 35
85. � YJ
7/. 2 25' •rl
•rl � .. .. +> 57.0 ]1 zo
12. B I�
2e.s IO
ao_
53
M � - - -,- -�---·- - ···,------r----
Y11
M y
�3 - - - - - - -
M Y"_2 _ __ __ _
M 2) _ __
o observed
0. 5 /. 0 /. f z.o
Centerline Deflection
Figure 19 . Load Versus Deflection at Midspan ,,. Beam TS-J
54
noticeable that lateral buckling of the flange was impending. At this
point , the vertical deflection was recorded as 2. 05 inches. An attempt
was made for another load interval , but at a load only slightly above
39 , 760 pounds , the compression flange buokled laterally, as shown in
Figure 20; the vertical deflection increased rapidly as the load
decreased. When the load dial was baok to zero, the reading on the
vertical deflection dial was 0 .492 inches .
Specimen TS-4
The third specimen to be tested under static load was beam TS-4.
Fourteen strain gages were mounted on this beam as shown in Figure 21.
Three of the strain gages were placed on the opposite side of the web of
this beam. Strain gage number 5 was located four inches from the top
flange , and gage number 10 was mo�nted at the neutral axis .
The first step was to load the beam to a moment of 41.838 ft-kips .
The rate of loading was 2 ,280 pound intervals , at the end of which ..
strain gage and deflection readings were taken. Final readings were
recorded at a moment of 41. 838 ft-kips and the beam was unloaded. At
the end of the loading and unloading of this first cycle the reading of
the vertical deflection dial was 0 . 000 inches .
The procedure for the second cycle was exactly like that used for
Specimen TS-J . The beam was first loaded to a moment o:f 6.1 .332 ft-kips
at intervals of 2, 280 pounds . This moment exceeded the yield limit of
the bottom. fibers of the �eb , M = 55 .263 ft-kips . Loading was con-Y1
tinued until the load on the beam. was greater than the yield moment of
the upper web fibers , M = 75 . 798 ft-kips . The beam was actually Y2
Figure 20. Lateral Buckling of Compression Flange of Specimen TS-3
55
3\
1
-2
--
.,, \ \
'\ \ - �7 \.
'- + '--9 · ''-H-- 5
--¥
_, -_11
12.
f
Figure 21. Strain Gage Arrangement Specimen TS-4
I
I I
\.n °'
. .
57
loaded to a moment of 87. 324 ft-kips. Before unloading, the horizontal
dials indicated some lateral movements ; however, after unloading, these
dials returned to their initial readings. The residual vertical deflec
tion after unloading the specimen was 0 . 199 inches.
A third cycle was tried with the purpose of reaching the theore
tical plastic moment limit. As with beam TS-3, the loading of the third
cycle coincided with the unloading of the second cycle, as is shown in
Figure 22 . At a load of 37,480 pounds, the lateral deflection became
quite pronounced. When the load exceeded 39, 760 pounds, the deflection
rate started increasing steadily ·and the lateral flange buckling
appeared as shown in Figure 23 . The highest load reached was 40, 900
pounds, which corresponded to a moment of 116. 565 ft-kips. This moment
was just above the theoretical plastic moment of 116. 068 fti--ki ps.
When the load dial read zero, the reading of the vertical deflection
dial was 0 . 622 inches. The flaking of the whitewash on the upper por
tion of the web in Zone A i_s shown in Figure 24.
..
12.8. 2
114.0
99.8
85. 5
(/)
71.2 •rl
·rl � "'
57,0 m (]) 0
0
42.8
28. ,
l+.-2.
o.o
4,
3S
10
�f!
2Q
15
I�
M flt- - - - - - - --- -- , -¼ - - - - - - - -
- M _1._ _ _ _ _ _ ___
end of
M Y2
. Oobserved
0. 5.� /.0 �-- 0 .
Centerline Deflection
Figure 22. Load Versus Deflection at Midspan Beam TS-4
58
Figure 23 . Lateral Buckling of Compression Flange of Specimen TS-4
59
60
Cl) E--l � Q)
•rl C) .. . Q) P-t
Cl)
4-1 0
� Q) � 0
N
� •rl
(/) Q) �
•rl H 'O rl Q)
•rl
rl m C)
•rl
$ .
Q) H &l
•rl Pr,..
. .
Specimen TF-1
CHAPTER IV
TEST RESULTS
A goal of 1,000,000 cycles of loading was established for the
fatfgue test of Specimen TF-1 . However, because of the relatively slow
rate of load application and frequent mechanical problems with the test
ing equipment, only 232,067 cycles of loading were actually achieved.
Investigation of the- stresses that were being developed in the
section under the 6,000 pound midspan load indicated that they were con
siderably less than the yield stresses of the three steels used in the
beam . However, this was not considered significant since a member may
fail after a certain number of load applications even if the maximum
stress in a single cycle is much less than the yield stress of the
material .
It was expected that high stresses in the vicinity of the welds
would develop which would reduce the permissible repeated range of
stress for failure . Fatigue failure was expected to initiate at a flaw
in the flange-to-web fillet weld and propagate through the flange and
into the web. However, daily checks of the deflection and of the weld
areas gave no indication of an impending failure or of a change in the
elastic properties of the beam. Since such a small number of load
applications was achieved, it was virtually impossible to draw any sig
nificant conclusions concerning the fatigue properties of tri-plate
beams.
62
Specimen HS-2
During the loading and unloading of the first cycle, the beam
behaved elastically following a straight line relationship on the ·
deflection versus load curve, Figure 16. The second cycle also indi
cated elastic behavior, and the observed deflection curve followed the
same path traced by the deflection points of the first cycle of loading.
However, the observed deflections for both cycles were less than the
theoretical deflection. This deviation may be attributed to some
restraint from the lateral supports. Although the late�al supports had
been heavily greased, it was noticed that some partial binding was
occurring between the lateral supports and the tension flange. This
problem. was easily corrected, and no further problems were encountered.
The observed load-deflection curve for the . third cycle· of loading
followed the path of the unloading of the second cycle. Although the
third cycle was not terminated until the moment had reached the oa.lcu
lated flange yield limit, the residual deflection after unloading this
cycle was relatively small. This residual deflection was 0.0011 of the
span length.
The load-deflection curve for this test indicates elastic
behavior far above the yield point stress in the web. Actually, the
deflection curve did not deviate appreoiably from. a straight line rela
tionship until a moment value in excess of the flange yield moment was
reached. Also, at this point, the observed deflection became greater
than the theoretical deflection.
The approximate analysis indicated a plastic moment of 115. 278
ft-kips for this beam.. When the theoretical plastic limit was reached,
63
distinct yield lines were observed on the upper web and flange; it was
evident that buckling of the compression flange was impending. Upon
increasing the load, the compression flapge buckled laterally in Zone A.
This yielding occurred in the region indicated by calculation, but at a
higher load. Assuming the calculated plastic moment to be a measure of
the ultimate strength, the specimen carried 5 percent more _moment than
calculated.
This test demonstrated that there is little adverse effect on the
ultimate carrying capacity of a hybrid . beam when the web of the member
is stressed beyond its yield strength. In fact, the load-deflecti•on
curve indicates elastic behavior throughout a wide load range. These
results compare favorably with the results of the previously mentioned
tests on the behavior of hybrid beams.
Specimen TS-3
During the first cycle of loading, the beam behaved elastically,
as expected. Since the frictional resistance of the heavily greased
late.ral supports had been virtually eliminated, the centerline vertical
deflection during this cycle was in very close agreement with the theo
retical deflection as shown in Figure 19. The initial loading of the
second cycle indicated a load-deflection curve coincident with that of
the first cycle. This coincidence was as expected, since this loading
was well within the etastic range.
The approximate analysis indicated first yielding would occur in
the lower web of Zone A at a moment of 55. 479 ft-kips. First flange
..
64
yield was calculated to occur in the compression flange of Zone A at a
moment of 94.567 ft-kips.
The first yield lines were observed in Zone A during the upper
limits of the second cycle of loading. Although the second cycle was
not terminated until the yield limits of b0th the bottom and t0p web
fibers had been exceeded, the only evidence of yielding was in the com
pression portion of the web. This yielding did not seem to influence
the behavior of the beam because no significant change in deflection
was evident in the load-deflection curve until after the yield limits
of the web material were exceeded.
During the loading of the third cycle, the beam behaved elasti
cally and followed a straight line pattern that very clesely coincided
with the unloading of the second cycle . This path deviated from the
straight line relationship when the loading reached the ·calculated yield
limit of the flanges. At this stage, flaking of the whitewash indicated
yielding in the compression flanges . There was little evidence ef yield
lines in the tension flange.
A slight buckling of the compression flange was visible at a
moment of 113. 316 ft-kips. At a moment of 115. 539 ft-kips, the compres
sion flange buckled laterally, representing 0. 7 percent deviatien from
the theoretical plastic moment of 116. 292 ft-kips. There was consider
able whitewash flaking, with particular concentrations in Zone A.
The load-deflecition curve, Figure 19, p0ints out that at loads
above the theoretical yield limits of the web, the observed deflections
are considerably greater than the theoretical deflections, r ,The centerline
,- , , .
deflection which was recorded just before failure represents an actual
deflection 15 percent greater than oalc�ated.
Specimen TS-4
The first cycle of loading on beam. TS-4 was completely elastic
without any residual deflection at the end of the cycle. As with beam.
TS-3, the observed centerline deflection of the first and second cycles
were coincident.
As the yield point of the web material was reached, the plot of
deflection versus load deviated from. a straight line, although no yield
ing was yet visible on the beam.. After this increase in deflection at
the upper stages of the second cycle, a straight line relationship
between load and deflection was again apparent until the m.om.ent reached
the yield limit of the flanges .
Unloading of the second cycle produced a residual deflection of
0 . 199 inches, which is equivalent to 0. 00087 of the span length. This
amount of residual deflection is relatively small ; where the dead load
co·mprises a major portion of the total - load, this residual can be
corrected by a very sm.all cambering of the beam .
In the upper stages of the third cycle the vertical centerline
deflection again showed a significant increase. Also, at this point the
lateral deflection became quite pronounced, and yield lines were
observed in the compression area of the specimen. Lateral buckling of
the compression flange did not take place, however, until the theoreti
cal plastic limit had been exceeded. This buckling occurred in Zone A,
as expected, but at a higher moment than indicated by the approximate
analysis.
66
The strain gages showed a uniform linear strain across the sec
tion with a negligible reading on strain gage number 10 which was
located at the neutral axis. When the flange buckled, the readings
were not consistent with any particular pattern.
The load versus deflection curve, • Figure 22 , indicates very close
agreement between the observed and the theoretical deflections at low
values of moment. However, as the ni.om.ent on the beam. is increased, the
actual deflections tend to becom.e · considerably greater than the calcu
lated values.
The load-deflection curves for all tests show that behavior of
the specimens was very nearly elastic throughout the entire load range.
The important fact brought out by these tests was that the member con
tinued to act elastically even after a considerable portion of the web
had been yielded.
All specimens in these tests had def'lections which were much
larger than the allowable live load deflection as specified by AASHO
(American Association of State Highway Officials) and AISC. These
deflections are not surprising since almost all of the load in these
tests could be considered as live load, which, as pointed out in the
theory, would result in an inadequate depth-to-span ratio. However,
this fact cannot necessarily be used as a measure of usefulness of the
member since, in actual practice, the live load will norm.ally be in
smaller proportion to the total load and deflections will be consider�
ably smaller.
The ability of the lateral bracing system. to prevent premature
lateral buckling was excellent. The rigidity of the lateral supports,
coupled . with the larger compression flange area, tended to resist .
lateral buckling in the tri-plate beams . Actually, this effect might be
com.pared to composite beams where the slab furnishes lateral support for
the compression flange . It has been suggested that the lateral buckling
problem. and the deflection problem encountered through the use of high-
t th t 1 fl b 1 d b . . t' t t . l0 s reng s ee anges can e reso ve y using composi e cons rue ion .
This method not only provides lateral support but also increases the
moment of inertia of the section so that deflections can be controlled .
Since theoretical analysis indicated that the shear would not
have any detrimental effect on reaching the plastic ·moment, it was
expected that the beams would fail by lateral buckling of the compres
sion flange . Although premature buckling was not a problem., lateral
buckling was the mode of failure in all · tests .
The high plastic moment value obtained in the test of Specimen
HS-2 can be attributed to strain-hardening . The four cycles of loading
applied in this test strained the beam through the plastic region far
in excess of the elastic limit strain . In such cas�s, strain hardening
occurs, and any further deformation will be accompanied by an increase
in the stress capacity of the material .
10 Toprac and Engler, p . 91 .
CHAPTER V
SUMMARY AND CONCLUSIONS
Summary
The results of the preceding tests a:re summarized in Table IV ,
which gives the calculated theoretical moment and the observed ultimate
moment :
Table IV
Theoretical Versus Observed Moment
Myf ' in M ' in M , in M * , in Test Y4 p 0
ft-kips ft-kips ft-kips ft-kips
HS-2 106. 888 115 . 278 121. 723
TS-J liJ. 104 116. 292 115. 539
TS-4 112. 906 116. 068 116. 565
*M is the observed plastic moment 0
Conclusions
From results gathered during the testing of the specimens, the
following conclusions were drawn :
1. The theoretical analysis appears to give a valid prediction
of the ultimate strength. The calculated values obtained
from equation (16 ) compared favorably with the test results.
2. When a tri-plate beam is loaded so that the yield points of
the upper and lower web are exceeded, a redistribution of
stress will occur and allow the beam. to continue to deform
elastically. As shown in Figures 19 and 22, the load
deflection curve deviated from a straight line relationship
when the yield points of the upper and lower web were
exceeded. However, upon further loading, a redistribution
of stress occurred, and the member continued to behave elas
tically until the yield points of the flanges were reached.
J. Deflection and lateral buckling become more important when
a tri-plate is used because of the reduction in flange
cross section. As previously pointed out, excessive deflec
tions were obtained in each test. Also, the mode of failure
for each beam was by lateral buckling of the compression
flange .
4. The use of tri-plate beams was shown to be feasible. The
beams tested developed an ultimate moment in the range of the
calculated plastic strength, M . . p
5 . The load carrying capacity of tri-plate beams is approxi-
mately equal to the ultimate strength of hybrid beams, as
seen from the tests. If there is a small price differential
between "T-1" steel and A441 steel, it means that the tri-plate
concept lends itself to a lighter and more economical section
than the hybrid beam.
6. The upper practical limit of bending strength for design pur
poses is the moment causing initial yielding in the tension
flange. Because it represents the moment above wliich the
. .
70
curvature deviates significantly from a straight line, elas
tic rather than plastic design limitations can be used.
Recommended Areas of Future Study
The study of the behavior of tri-plate beams is relatively new
and should be investigated more extensively in the following areas :
L A more complete study of the fatigue properties of tri-plate
beams should be undertaken 8
2. Tri-plate beams with thin webs should be investigated with
particular emphasis on the utilization and study of
stiffeners.
J o The use of tri-plate beams in comp0site construction should
be studied.
4. The shifting of the neutral axis in tri-plate beams, and the
effect on the bending behavior, should be extensively ana
lyzed o
..
NOTATION
The following notation is used in this thesis:
A= cross sectional area of the entire beam
A = cross sectional area of the web w
bb - bottom flange width
bt = top flange width
C = web depth
cb = distance from the neutral axis to the extreme bottom. web
fibers
71
ct = distance from. the neutral axis to the extreme top web fibers
d - beam depth
¾ = distance from. the neutral axis to the extreme bottom. flange
fibers
dt = distance from the neutral axis to the extreme top flange
fibers
E = modulus of elasti9ity of steel
I = moment of inertia of the entire cross section of the beam
1fb = moment of inertia of the bottom. flange
1ft = moment of inertia of the top flange
Kl = constant of end restraint
K2 = constant of end restraint
L = span length
M = bending moment
M = observed bending moment
M - plastic bending moment of the entire beam cross section
..
M = bending moment at the initial yielding in the web (hybrid yw
beam)
72
Myf = bending moment at the initial yielding in the flanges (hybrid
beam)
M = bending moment at the initial yielding of the lower web Y1
fibers (upper limit Stage I)
M = bending moment at the initial yielding of the upper web Yz
fibers (upper limit Stage II)
M = bending moment at the initial yielding of the upper flange Y3
(upper limit Stage III) ·
M = bending moment at the initial yielding of the lower flange Y4
(upper limit Stage IV)
P = concentrated load
tf = flange thickness
t = web thickness
V = shear force
V = plastic shear of the web p
wl = uniformly distributed live load
wT
= total uniformly distributed load
¾= distance from left support to Zone B
x = distance from left support to Zone C C
y = distance from. the neutral axis to the plane of stresses
Z = plastic modulus of the entire beam cross section
Zfb = plastic modulus of the bottom. flange
Zft = plastic modulus of the top flange
Z ·- plastic modulus of the web w
f.i = centerline deflection
E: - strain
¢=curvature
er= stress at the extreme fibers of the section
o;. = yield stresses at the extreme fibers of the
cry - yield stress of the upper flange
Cy4 = yield stress of the lower flange
r:ryw = yield stress of the web
P = radius of curvature
, = shear stress in the web
'y = shear yield stress of the web
w = total jack load
73
section
1 .
2 .
3 .
4.
5.
6.
7.
8 .
9.
10.
11 .
12 .,
BIBLIOGRAPHY
Haaijer, G . , "Economy of High Strength Steel Structural Members f " Transactions, American Society of Civil Engineers, vol. 12.S , 1963 ..
Toprac, A ,. A . , and R. A . Engler, " Plate Girders With High-Strength Steel Flanges and Carbon Steel Webs, " Proceedings, American Institute of Steel Construction, 1961.
Frost, Ronald W . , and C. G . Schilling, "Behavior o.f Hybrid Beams Subjected to Static Loads," Journal of the Structural Division, ASCE, vol. 90, no . ST3, Proceedings Paper 3928, June, 1964 .
Sarsam., Jamal B . , "Behavior of Thin Web Hybrid Beams Subjected To Static Loads," Master of Science Thesis, South Dakota State University, Brookings, South Dakota, August, 1965 .
Joint Cemm.ittee of Welding Research Council and the American Society of Civil Engineers, Cbmm.entary on Plastic Design in Steel, 1961 .
Beedle, Lynn S . , and Associates, Structural Steel Design, The Ronald Press Company, New York, 1964.
Gilligan, J . A . , "Higher Strength Structural Steels, " Civil Engineering, vol. 34, no . 11, November, 1964.
Beedle, Lynn S . , Plastic Design of Steel Fram.es, John Wiley and Sons, Inc . , New York, 1964 .
Basler, K . , " Strength of Plate Girders in Shear, " Transacti0ns, · American Society of Civil Engineers, vol. 128, 1963.
Reneker, W . D. , and C . E . Ekberg, "Flexural Fatigue Tests of Prestressed Steel I-Beams, " Journal of the Structural Division, ASCE, vol . 90, no . ST2, Proceedings Paper 3868, April, 1964 .
Basler, K. , "Strength of . Plate Girders Under Combined Bending and Shear, " Transactions, American Society of Civil Engineers, vol . 128, 1963 .
Stuessi, F . , "Theory and Test Results on the .Fatigue of Metals," Transactions, American Society of Civil Engineers, voL 128, 1963 .
..
APPENDIX A
DERIVATION OF MOMENT-CURVATURE RELATIONSHIPS
FOR PURE BENDING
The moment-curvature relationship at any given stage of loading
can be obtained by integrating the stress areas ac cording to the general
expression
M � E� (Ift + Ifb)
where
Therefore
M =
Yo - �
E�
E� (Ift + 1fb)
M = {Q"dAy J lrea
twcb 2
+
0-yw 2
t Cb 2
+ rr w yw 2
+ ·
+
t 2 t 2
r:::T ifo
+ 2 � ifo
yw 2 yw 3
er. tv7 0
2
yw 6
t C7: 3 w YJl.
6E2¢2 (5)
..
77
Stage III
where
and - � Yo - E�
Therefore
( 8 )
Stage IV
where
.,
Therefore
3 tw �
yw M = 'J: zf t + E(J)'Ifb + V:: z - 2 2 Y3 yw w JE �
Stage VI
78
(11)
M = f
;
t
b dy . y + f;\bdy• y +}:
b
t dy• y + f
;
t
t dy• y YJ t y4 yw w yw w
t b Yo o
d 2 2 <\ 2 2 t 0- 3 c� ) � ) = V- bt ( � +
V-y /b ( 2 + r;;r ywzw
w yw Y3 JE2¢2
t v:: 3
= r;r_ 2ft + tr 2fb + V" z. w '::DI.. (14)
Y3 Y4 yw w JE2�2
For sections which only have symmetry about an axis in the plane
of bending, the position of the neutral axis at any stage of loading
must be such that equilibrium. of horizontal forces is maintained. Thus,
the neutral axis of a tri-plate beam actually shifts as the stresses on
the beam are increased. However, because of the particular nature of
the tri-plate beam cross section used in this . study, this shifting of
the neutral axis was determined to be very small . It was assumed that
such a minimal movement of the neutral axis would not appreciably
influence the theoretical calculations . For the purpose of this study,
the theoretical analysis is thus based on a stationary neutral axis.
APPENDIX B
DERIVATION OF THEORETICAL DEFLECTION EQUATION
•
p p
3PL SEI
80
p p
L
L/5 �--..;....;;;..___.L...-___ ____.� ___ ____._ _______________ _.., M
From which
and
� • = lllPLJ
J000EI
Therefore
� = 12. PLJ lllPLJ
2 9 5EI - J000EI
EI
d
( 20 )
APPENDIX C
PUMP LOAD , ACTUAL LOAD, AND ACTUAL MOMENT RELATIONSHIP
FOR EACH SPECIMEN
.,
82
Pump Readings , Actual Load , and Actual Moment Relationship
Beam. HS-2 Beam TS-J Beam. TS-4 Dial . w M w M w M
Readings lbs ft-lbs lbs ft-lbs lbs ft-lbs
0 1 , 100 3 , 135 1 , 000 2 , 850 1 , 000 2·, 850
2 3 , 380 9 , 633 3 , 280 9 , 348 3 , 280 9 , 348
4 5 ,660 16 , 131 5,560 15 , 846 5 ,560 15 , 846
6 7 , 940 22 , 629 7 , 840 22 , 344 7 , 840 22 , 344
8 10 , 220 29 , 127 10 , 120 28 , 842 10 , 120 28 , 842
10 12 , 500 35 , 625 12 , 400 35 , 340 12 , 400 35 , 340
12 14 , 780 42 , 123 14 , 680 41 , 838 14 , 680 41 , 838
14 17 , 060 48 , 621 16 , 960 48 , 336 16 , 960 48 , 336
16 19 , 340 55 , 119 19 , 240 54 , 834 19 , 240 54 , 834
18 21 , 620 61 , 617 21 , 520 61 , 332 21 , 520 61 , 332
20 23 , 900 r-68 , 115 23 , 800 67 , 830 23 , 800 67 , 830
22 26 , 180 74 , 613 26 , 080 74 , 328 26 , 080 74 , 328
24 28 , 460 81 , 111 28 , 360 80 , 826 28 , 360 80 , 826
26 30 , 740 87 , 609 30 , 640 87 , 324 30 , 640 87 , 324
28 33 , 020 94 , 107 32 , 920 93 , 822 32 , 920 93 , 822
30 35 , 300 100 , 605 35 , 200 100 , 320 35 , 200 100 , 320
32 37 , 580 107 , 103 37 , 480 106 , 818 37 , 480 106 , 818
34 39 , 860 113 , 601 39 , 760 113 , 316 39 , 760 113 , 316
36 42 , 140 120 , 099 42 , 040 119 , 814 42 , 040 119 , 814
38 44 , 420 126 , 597 44 , 320 126 , 312 44 , 320 126 , 312
40 46 , 700 133 , 095 46 , 600 132 , 810 46 , 600 132, 810